A detailed comparison of continuum and valence force field strain calculations in quantum-dot structures is presented with particular emphasis to boundary conditions, their implementation in the finite-element method, and associated implications for electronic states. The first part of this work provides the equation framework for the elastic continuum model including piezoelectric effects in crystal structures as well as detailing the Keating model equations used in the atomistic valence force field calculations. Given the variety of possible structure shapes, a choice of pyramidal, spherical and cubic-dot shapes is made having in mind their pronounced shape differences and practical relevance. In this part boundary conditions are also considered; in particular the relevance of imposing different types of boundary conditions is highlighted and discussed.
We present a study of blue III-nitride light-emitting diodes (LEDs) with multiple quantum well (MQW) and quantum dot (QD) active regions (ARs), comparing experimental and theoretical results. The LED samples were grown by metalorganic vapor phase epitaxy, utilizing growth interruption in the hydrogen/nitrogen atmosphere and variable reactor pressure to control the AR microstructure. Realistic configuration of the QD AR implied in simulations was directly extracted from HRTEM characterization of the grown QD-based structures. Multi-scale 2D simulations of the carrier transport inside the multiple QD AR have revealed a non-trivial pathway for carrier injection into the dots. Electrons and holes are found to penetrate deep into the multi-layer AR through the gaps between individual QDs and get into the dots via their side edges rather than via top and bottom interfaces. This enables a more homogeneous carrier distribution among the dots situated in different layers than among the laterally uniform quantum well (QWs) in the MQW AR. As a result, a lower turn-on voltage is predicted for QD-based LEDs, as compared to MQW ones. Simulations did not show any remarkable difference in the efficiencies of the MQW and QD-based LEDs, if the same recombination coefficients are utilized, i.e. a similar crystal quality of both types of LED structures is assumed. Measurements of the current-voltage characteristics of LEDs with both kinds of the AR have shown their close similarity, in contrast to theoretical predictions. This implies the conventional assumption of laterally uniform QWs not to be likely an adequate approximation for the carrier transport in MQW LED structures. Optical characterization of MQW and QD-based LEDs has demonstrated that the later ones exhibit a higher efficiency, which could be attributed to better crystal quality of the grown QD-based structures. The difference in the crystal quality explains the recently observed correlation between the growth pressure of LED structures and their efficiency and should be taken into account while further comparing performances of MQW and QD-based LEDs. In contrast to experimental results, our simulations did not reveal any advantages of using QD-based ARs over the MQW ones, if the same recombination constants are assumed for both cases. This fact demonstrates importance of accounting for growth-dependent factors, like crystal quality, which may limit the device performance. Nevertheless, a more uniform carrier injection into multi-layer QD ARs predicted by modeling may serve as the basis for further improvement of LED efficiency by lowering carrier density in individual QDs and, hence, suppressing the Auger recombination losses.
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